**Mathematical Classification and Rheological Properties of Ring Catenane
Structures**

K Hagita and T Murashima and N Sakata, MACROMOLECULES (2021).

DOI: 10.1021/acs.macromol.1c01705

The rheological properties of polycatenanes were investigated by coarse-
grained molecular dynamics simulations using the Kremer-Grest-type bead-
spring model. To prevent the combination number from explosively
increasing, systematic structural models of **n**catenanes (n = 2, 3, and
4) were generated using a mathematical graph representation. It was
confirmed that the behavior of the storage and loss moduli, G'(omega)
and G ''(omega), respectively, depends approximately on the number of
beads (monomer units) per catenane at low frequencies. We found that the
crossing numbers affected the behaviors of G'(omega) and G ''(omega) in
the immediate frequency range. Moreover, the storage modulus at the
middle frequency tends to behave as a linear function of the crossing
number. For the small rings, an exhaustive study based on mathematics
revealed that even if the crossing number is the same, there are cases
where the storage modulus at the middle frequency becomes exceptionally
large due to the difference in linking rings. For the smaller ring sizes
and/or larger crossing numbers, we discovered a sol-gel transition in
the G'(omega) and G ''(omega) plots. For the Kremer-Grest model of the
peptide **6**catenane and peptide **4**catenane structures that have been
experimentally prepared by the Fujita group, the threshold ring-size
value for this transition was found to be approximately 25 and 23,
respectively.

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